IC/97/14

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

REGIONAL STRUCTURE MODELLING AND SOURCE INVERSION FOR THE 1992 EARTHQUAKE

H. Dufumier1 Dipartimento di Scienze della Terra, via Weiss 1, Trieste, Italy,

A. Michelini2 Dipartimento di Scienze della Terra, via Weiss 1, Trieste, Italy,

Z. Du International Centre for Theoretical Physics, SAND Group, Trieste, Italy and Dipartimento di Scienze della Terra, via Weiss 1, Trieste, Italy,

I. Bondar International Centre for Theoretical Physics. SAND Group, Trieste, Italy and Seismological Observatory, Hungarian Academy of Sciences, Meredek u. 18, 1112 Budapest, Hungary,

J, Sileny Geophysical Institute, Academy of Sciences of the , Bocni 11/1401, 141 31-Praha 4, Czech Republic,

W. Mao Department of Earth Sciences, University of Leeds. Leeds LS2 9JT. United Kingdom,

S. Kravanja Dipartimento di Scienze della Terra, via Weiss 1, Trieste, Italy

and

G.F. Panza International Centre for Theoretical Physics, SAND Group, Trieste, Italy and Dipartimento di Scienze della Terra, via Weiss 1. Trieste, Italy.

MIRAMARE - TRIESTE February 1997

1Present address: Institut de Physique du Globe, 5 rue Descartes, 67084 Strasbourg Cedex, . 2Prcsent address: Osservatorio Geofisico Sperimentale, P.O. Box 2011, Opicina, Italy. Abstract: The Mw = 5.4 Roermond earthquake of April 13, 1992, is used as a "test" earthquake for the development of source inversion methods at a regional scale in Europe. We combine structural modelling of the European continent (Du et al., 1997) with two source inversion methods derived from Sileny et al. (1992), and Mao et al. (1994). We show that following this strategy, it is possible to fully analyze the inverse problem of the hypocentral relocation, source mechanism and rupture history. We define and discuss our methodology on the basis of the inverse problem and of the associated tools. The results of our application to the Roermond earthquake are discussed at the light of other previously published solutions. Such an approach appears to offer a promising tool for the global description of seismic sources in regions well studied from the structural point of view, through waveform inversion of a few regional records.

Key Words: Roermond, Source Inversion, Regional structure.

Introduction

The April 13, 1992, Mw = 5.4 Roermond earthquake, in the Roer Valley (The ), belongs to the largest earthquakes which occurred in North-Western Europe in this century. It has already been extensively studied with different approaches. We refer to Geluk et al (1994), van den Berg (1994) and Trifonov et al. (1994) for the geological aspects, to Camelbeeck and van Eck (1994) and Camelbeeck et al. (1994) for the seismological studies of the main event and of its aftershocks, and more generally to the special issue published about this earthquake by van Eck and Davenport (1994) for an extensive overview of seismic hazard, tectonic, seismological, engineering and hydrogeological aspects. Concerning the source inversions for this earthquake, we note the CMT solution at a global scale (Dziewonski et al., 1993), and the regional scale inversion by Braumiller et al. (1994). Loohuis and van Eck (1996) also presented a joint inversion for the regional source mechanisms and stress tensor. This earthquake is used as a 'test' earthquake to improve new developments in regional structure modelling and source inversion at the European scale. In fact developments in these two scismological fields should not be disconnected, since source retrieval highly depends on path effects at the regional scale, as at the global scale (Dufumier and Trampert, 1997). We combine structural modelling, derived from the regional I-data set for the European continent (Du et al., 1997) with two source inversion methods derived from local-scale methods (Sileny and Panza, 1991; Mao el al., 1994). In particular, we consider waveform inversions in time domain of the vertical component of motion.

/. Structural modelling

Regional studies require accurate modelling of the geological structures in the region of interest. We show in figure 1 the six regional paths for which broadband data were easily available on line in the first years following the event, and used in this study. Available data from other stations too close to these ones were not considered, since they would not bring independant information (cf Dufumier and Cara, 1995) and lead to undesirable data redundancy effects (Michelini, 1997). We use the three-dimensional I-data set model of Europe (Du el al., 1997) to obtain cross- sections of the crustal and lithospheric structures along these paths. The I-data set of Europe is a 3 - D structural model of the tcctosphcrc. This model has been assembled from the published literature and it includes all the principal geological and tectonic features that have been recognized on the regional scale. The I-data set is made out of approximately 6000 1-D structures and linear interpolation is used throughout to find P-wave and S-wave velocities, density and attenuation within the structure. As illustration, we present in figure 2 our most heterogeneous cross-section, corresponding to the great circle path Roermond - ESK. Overall, the I-data set has been found to provide reliable dispersion measurements on the whole Euro-Mediterranean domain for periods greater than 15 seconds (Du et a!., 1997). However, to obtain detailed information on the source mechanism, it is necessary to model the wavcficld, i.e. to determine the Green's functions, at shorter periods. Because existing methods for forward modelling in 3-D or across 2-D cross-section (e.g. finite differences, ray methods or boundary integrals) arc either computationally expensive, or require decomposition of the wavefield, we follow the simpler approach of averaging the 3-D I-data set model into 1 -D structures along each source-receiver path, and forward modelling is performed using a modal summation method (e.g. Panza, 1985). If the 2-D cross-section of the I-data set model would have displayed strong discontinuities, it would have been necessary to use 2-D modal summation methods, including transmission and coupling at the interfaces (Vaccari et ai., 1989), but this is not the case for the region we examine here. Averaging, however, may remove a significant part of the original 3-D information and, therefore, it should be performed with caution, to preserve the information pertinent for source inversion. We considered here two types of averaging: - the first one, called "layer-averaging", preserves the layer discontinuities through interpolation within crustal and mantle layers separately, averaging the depths of the main interfaces. - the second one, called "depth-averaging", averages the velocities at each depth along the source-receiver path, and results into a more continuous model. The 1 -D models obtained using the two types of averaging arc presented in figure 3 for the six paths of figure 1. We also show the model for the source region as determined from the I-data set. This model is consistent with those published in the literature (e.g. Trifonov et ai., 1994). The effects of averaging on the source inversion will be studied in the next part; but it can already be noted that it affects significantly the dispersion of the first modes (figure 4a), and, therefore, the synthetic waveforms used in the inversion (figure 4c). To ensure that both data and synthetics have a similar time-frequency content, we apply to both sets a variable-period velocity filter (Levshin et ai., 1972; Cara, 1973). The filtering limits are fixed from the double observation of the data spectrogram (or multiple time-frequency analysis, sec Kocaoglu and Long, 1993, for a review of the techniques) and of the dispersion curves associated to the structure. An example is shown on figure 4, for the path Rocmnond-WET. In figure 4a we present the synthetic dispersion curves, according to the two averaging methods; while in figure 4b we show the spectrogram of the observed data (following Levshin el ai., 1992), with the selected filtering window. Figure 4c shows the effect of the variable filter on the original data and on corresponding synthetic seismograms, considering the two types of averaged structures. It can be seen that the use of an appropriate filtering window can help to adjust the time-frequency content of the synthetic seismograms to the data one without removing information from the original waveform. We also use similar filtering windows for the other paths. In addition, a common low pass filtering is applied to all the seismograms, defining the lowest period to be used in the inversion. This period can be adjusted, depending on the preference given to the resolution of the source model or to the quality of data fitting. We used cutting periods of 1 to 10 seconds, the most significative results shown here corresponding to cutting periods around 3 to 5 seconds. Thirty modes arc used to fit the time-frequency content of the data, so that we achieve the complete theoretical modelling of the seismograms from the S-wave to the end of the direct Raylcigh wave. The major part of the signal is kept in the inversion windows, corresponding to the zone of good signal to noise ratio that can be satisfactorily fitted using the summation of the most energetic modes. Anyhow, the data of the station HAM were not kept in short-period inversions, because the local influence of the Northern sedimentary basin could not be correctly taken into account from short-period derivations of the I-data set model. The methodology of data processing developed here will now allow us to perform waveform inversions of complete seismograms at a regional scale, where information on source and structures usually strongly mix together.

//. Extended Monte-Carlo search

In our first source inversion method, we want to consider the possible trade-offs between source and structural parameters, without constraining the solution with a priori assumptions. To this purpose, we use a method previously developed by Sileny (Sileny and Panza, 1991; Sileny et al., 1992; Campus et al., 1996) for local moderate-size events. Thanks to the "ovcrparamctrization" of the source process by means of independant moment tensor rate functions, the method has a capacity to absorb partly the effects of inaccurate Green functions modelling due to a lack of detailed information on the medium. Additional degrees of freedom enhancing this capacity are the freedom of the hypoccntrc to move inside a large predefined volume and the freedom to "mix" two extreme structural models of the area under study to get synthetics which yield the best fit with the observed waveforms. In detail, we explore a 4-D space, three dimensions of which arc the hypocentral coordinates and the last one describes the variation between two end structures. This exploration is performed using a Monte-Carlo random search combined with a Simplex minimization. In practice, at each generic point of the 4-D space, we perform a linear inversion for the six moment rate functions M-.(t), so that the moment tensor is allowed to vary freely through the predefined time window. Before describing the results of the 4-D search, we point out that a first limitation of this approach derives from the ill-conditioning which results from the very large number of parameters inverted simultaneously {i.e. at each point of the 4-D space, 6 times the number of triangles required to describe the source history). The condition number (ratio between the extreme singular values of the theory matrix) for a 60-unknowns problem (10 triangles for each M ,-j component) is about 3000 in a large epicentral zone inside the actual station distribution, and it decreases by a factor 10 when using two times less parameters. It is greatly stabilized by the station ESK which opens the station network, but suffers from a non regular repartition of the data {e.g. Michelini, 1997). For a given station distribution, the conditionning can be improved by reducing the number of parameters. For example, by eliminating the isotropic component of the mechanism when it is supposed to be ncgligcablc or spurious. The paramctrization of the moment rate functions by a set of overlapping triangles can also be revised considering the resolution of each element M;j{t) observed from the resolution matrix (Tarantola, 1987), for example by constraining or reparameterizing the part of the source history which would appear to be poorly solved. We now use a paramctrization by five deviatoric moment rate functions made up of 6 triangles (30 parameters). By introducing a degree of freedom for the structure and leaving the parameters free during the exploration of the 4-D space, we do not claim to perform an optimization of the structure. This just allows more flexibility in the specification of the Green's functions, between two relatively close end models, as discussed by Campus et al. (1996). Anyhow, an appropriate choice of the two bounding structures may help to illustrate the close links between source inversion and structure modelling. We first present in figure 5 the minimization of the misfit function at a cut-off period of 3 seconds, varying the source depth and the structural parameter Y which ranges between 0 and 1. Y=0 represents a unique, vertically layered, homogeneous structure for the whole region (the source structure of figure 3). Y=l represents particular structures for each source-station path: in figure 5a, the laycr-avcraged structures of figure 3, and in figure 5b the depth-avcraged ones. The extension of the search grid for the source depth is from 8 to 28 km, centered around depths published in previous studies (14.6 to 21 km, sec Table 1). Considering the behaviour of the misfit function near the extremities of the Y axis, it appears that a unique homogeneous structure is to be preferred to the various layer-averaged structures, whereas the depth-averaged structures should be preferred to the homogeneous one. It should be noted that these preferences do not change at longer periods (10 sec), or when removing from the inversion the most heterogeneous path Roermond-ESK. This suggests that the needs of source inversion (a global vision of the source which should not depend on the paths) may go against geological realities (the prevalence of heterogeneity and of discontinuous structures). Indeed the inverse problem has a smoothing effect, connected with the wavelengths used, which may explain that if some heterogeneity is needed, it is preferred to be smooth and continuous. The preferred source depth is affected by the choice of the structural model: it is around 14 + 2 km for the unique homogeneous structure and 18 ±2 km for the depth-averaged heterogeneous one. Since the amplitudes of the minima arc about the same, it is difficult to prefer one model to the other one, but this illustrates how much the "best" depth may depend on the way a 3-D medium is mapped into a 1-D model, and highlight the importance of appropriate structural modelling in source inversion problems. For the next steps the depth-averaged structures (different for each source-station path) have been adopted.

The other dimensions of the space of parameters to be explored are the epicentral latitude and longitude. We present in figure 6a the results of the minimization of the misfit function at 3 seconds over this space. The exploration grid is centered around locations previously published for this event and extends from 51.00° to 51.30° North and 5.75° to 6.05° East {i.e. the hypoccntrc is allowed to move within an approximately 30x20x20 km volume). The previously published locations arc grouped around latitudes of 51.150° to 51.170°N, while the longitudes arc not as robust, ranging from 5.798° to 5.970°E (Table 1). Note that we do not take into account the ccntroid location (Dzicwonski el al., 1993), being too far from the other studies (some 50 km to the North-West). Our inversion at 3 seconds presents two non significant local minima and a clear global minimum around 51.207°N, 5.952°E, having a visual extension of about 3 km. The longitude is clearly constrained to be in the eastern part of the grid, well in agreement with the other regional studies, while the latitude is slightly more to the North. It is interesting to compare this inversion with the inversions at longer periods, such as the 10 seconds one presented in figure 6b. If the two locations arc consistent with the regional ones, and although the misfit to filtered data is obviously much lower than the short-period one, the relocation at 10 seconds is not significant due to its lack of precision: the computed standard error on the epicentral location is about one fourth of the wavelength, i.e. ±3 km at 3 sec and ±10 km at 10 sec. At shorter periods, the misfit function becomes noisy, with many local minima and a very poor minimization. Complementary synthetic tests have confirmed this analysis of the non linear problem and show that getting a clear unique minimum may not be guaranteed below 5 seconds at similar regional distances (see also figure 10). The fact that our present minimum extends over the epicentral grid along a direction rather similar to the extension of the Roer Valley Graben does not imply that we have "mapped" the . It probably reflects cither the acquisition geometry, or the undctcrminacy in the source location in the point source approximation, several nearby nucleation points being able to describe cquivalcntly the scismograms at a period close to the source duration. It may also explain the larger discrepancy observed in previously published longitudes, comparatively to the latitudes.

Since we are not interested in a possible isotropic component for this tectonic event, and since its interpretation should be subject to due caution (Dufumicr and Rivera, 1997), we performed here only the inversion for deviatoric moment rate functions. Some tests have, in fact, confirmed that the volumetric components obtained for this event in full moment rate inversions were not physically dominant but, rather, artefacts. To this purpose, we developed a control of the elastic ratio )J\i associated to the tensor My- at each time step (following Dufumicr and Rivera, 1997), which helps to discriminate between physical and spurious volumetric components associated to a classical (but non unique) tectonic model of a fault plane with arbitrary slip. In the case the study of the isotropic part would be desirable, its interpretation in terms of tectonic and non-tectonic volumetric parts would be necessary (e.g. Rouland el al., 1997). No other constraints (apart a slight damping ratio of 0.001) were applied in this first explorative part of the inversion, in order to keep the solution as free as possible. In figure 7 we show the fit, at 3 seconds, between the data and the synthetics obtained from this unconstrained inversion at the best point of the 4-D space of the parameters, according to the criteria of misfit minimization. The minimum value of the misfit function is 0.73 (a value of 1 indicates no minimization). At 5 seconds, the minimal misfit would be 0.59, and 0.31 at 10 seconds, but then the description of the model parameters starts to be imprecise. As an alternative, standard, measure of the data fitting, the normalized correlations between the data and the synthetics are respectively of 0.65, 0.72 and 0.91 at 3, 5 and 10 seconds. In a second step, following the scheme of Silcny el at. (1992), the average source mechanism is then calculated by factorization of the best moment rate functions:

Mu{t) -^My.MJt)

Since the moment rate functions were allowed to vary freely during the first part of the inversion scheme, a second constraint has to be applied a posteriori to discard possible artefacts which may appear in the source history, and in particular backslips. The positivity of the source time function Mjt) implies, in practice, that we factorize only one "side" of the moment rate functions. The resulting undeterminacy on the sign of the moment tensor My can be solved when it is recognized that it should be correlated with some clear first-motion polarities. We obtain the position of the polarities on the focal sphere using a ray-tracing method for the structures that we have used, and we then compute the correlation coefficient between the polarities and the P-wave radiation. The best correlation and the RMS of the factorization determine the choice of the average moment tensor in case of a sign indeterminacy.

In absence of any a priori constraint imposed to the My(t), the average moment tensor M//; i.e. the correlated part of the moment rate functions, may remain a small part of the complete moment rates. For example, in the free inversion of figure 8a, the RMS error between the factored and original moment rate functions is only of 0.617 and the seismic moment Mo of the constant average tensor, 5.6x 1017 Nm, although high, represents only 54% of the total unconstrained moment release. The uncorrelated part was kept in the first part of the inversion, with the idea that it may integrate errors in structural modelling and noise effects, and it is removed at this second stage. But, although the global radiation of the average moment tensor obtained here at 3 seconds (figure 8a) is coherent with the published solutions for this earthquake (figure 9), the best double- couple is rather unstable in our inversions, with a strike component poorly controlled, and the large amount of reversed backslip component obtained may appear exagerated. This may result from the fact that the final mechanism comes from the a posteriori simplification of a much more complex model, which faces the danger of ill-conditionning due to the large number of parameters. Longer periods (15 s and more) would provide us with a direct linear estimation of the moment tensor (e.g. Kanamori and Given, 1981; Dufumier and Cara, 1995), but loosing then the information on the epicentral relocation and on the source history. Applying more damping to the original, poorly conditionned, system, will also stabilize, artificially, the inversion. For example, we show in figure 8b the result of the same inversion as before when applying a rather strong damping ratio of 0.01 (i.e. the damping is of the order of 10% of the largest values of the theory matrix): the source time function becomes more unilateral and continuous, and the non-correlated part of the moment rate functions is concentrated at the beginning. The average tensor and its seismic moment (56% of the total moment release, but almost all the moment release after the theoretical origin time) are more in agreement with other published solutions. On the other hand, the fit to the data deteriorates (the correlation is of 0.62). And the use of a strong damping affects artificially and globally the description of the model, while the methodology originally proposed by Sileny et at. (1992), and discussed by Campus et at. (1996), is expected to absorb the inaccuracies in the structural modelling and allows for a great flexibility in the retrieval of the model. In future developments, a less constraining way of limiting backslip in the moment rate functions might be to keep a low global constraint and introduce some correlation between the different time steps Ml}(i) of the moment rate functions, but leaving the geometrical part free.

In summary, the method, performing a detailed exploration of the hypoccntral space, offers a good overview of the non linearity of the global problem and provides a reliable hypocentral location together with an identification of the global minimum. It offers flexibility and freedom in the retrieval of a detailed source model. Aiming now to obtain rapidly a constrained simple description of the source, we will use the free model previously obtained as a starting model, and explore locally the solution using a gradient method.

///. Double-couple solution

The second method used in this study follows Mao's approach (Mao et al., 1994; Nformi et al., 1996). It inverts for the hypocentral relocation, the best double-couple and the source history, using a gradient method. The original method has been modified for application at the regional scale, by replacing the inversion for plane epicentral adjustments by that for the epicentral latitude and longitude. The method of the weighting matrix, introduced to compensate for the scaling discrepancies between the different classes of physical parameters, has been also replaced by the method of separation of parameters (Spencer and Gubbins, 1980; Mao and Dufumicr, 1997). The global system is inverted in three successive steps, subdividing it into three sub-systems: mechanism, source time function and hypoccntrc. As major improvements we obtain the non- penalisation of one set of parameters when another sub-system is poorly conditioned, and the possibility to use different damping factors to control the inversion of the various sets of parameters. The number of parameters to solve is much lower than in the previous method (Sileny et al., 1992): strike, dip, rake, 3 hypocentral coordinates and N overlapping triangles for the description of the source time function. However and because it is a linearized iterative inversion, it requires an a priori initial model and it will converge locally, to the closest minimum. Therefore the solution can be unstable and initial-model dependent at high frequencies, when local minima arc unidentified, numerous and shallow.

In order to investigate the relevance of the initial value of the various parameters as the method converges towards the true minimum, we performed some synthetic tests, using the configuration of the available stations (figure 1). The first condition is on the initial mechanism. Although it is not possible to give numerical margins for the strike, dip and rake angles (because of their non-linear and cyclic behaviour), it was found that the method converges to the true mechanism whenever the initial double-couple describes the same type of mechanism (normal fault, reverse fault, or strike-slip) as the true one. To this end, we have used the preliminary unconstrained solution obtained from Silcny's method (figure 8a). The second subset of parameters requires an initial estimate of the hypoccntcr, which should never be too far apart from the "true" one in order to retrieve it. In particular, synthetic tests performed at periods of 0.1, 1, 5 and 10 seconds, show that the largest possible distance of the initial hypocentre from the true one is approximately proportional to the wavelength. The method converges correctly when the original epicentre is within + 1.2 wavelength from the true one, while the initial depth can be up to ± 2.3 wavelengths from the true one (figure 10). Therefore, considering our a priori knowledge of the source location, we should not experience multiple local convergences when working at periods greater than 2 seconds. In practice, we will start our real data analysis using periods of 5 sec and longer, to obtain a clear and rapid convergence. It is interesting to compare the previous convergence limits with the final standard errors obtained on the model in section //, which arc also relatively proportional to the wavelength: the ellipse of error shows a great axis of ±0.25 wavelength for the epicentre and ±0.15 wavelength for the depth (figure 10). In more concrete terms, we have a precision of + 3 km on the depth and + 5 km on the epicentre at 5 seconds. The different alignment of the ellipses of convergence and of error show a relatively poor resolving power for the cpiccntral relocation, and a better resolution of the depth. These values depend, of course, on the station distribution and will improve using a more homogeneous distribution, but the present realistic case already shows that most of the source process can be, however, described using a sparse network. The minimal station configuration required is in fact 4 vertical-component stations providing an azimuthal aperture of 70° (figure 11). For distributions having a narrower aperture, even the 3 parameters of a double-couple cannot be satisfactorily solved. Note also the difference of conditioning between the current inversion for a double-couple and a source time function made of 10 triangles (condition numbers lower than 10 for each sub-system) and the corresponding one for the inversion of a time-variable full moment tensor (about 3000 for the 60 parameters describing the mechanism and the source history, c/part

We present in figure 12 the results of the real-data inversion at 5 seconds for different iterations. The mechanism, the source time function (parametrized by 5 overlapping triangles) and the hypoccntcr converge progressively to a more stable solution. The (normalized) RMS reduction is quite remarkable, passing from 0.52 to 0.24, whereas the average correlation between the data and the synthetics increases from 0.33 to 0.57. Note that the correlation coefficient is a mesure of the fit common to the two inversion methods described here and therefore the latest values can be compared with the correlation of 0.72 obtained at 5 seconds with Silcny's method, which involves many more degrees of freedom. The focal mechanism is now fully in agreement with other published solutions (figure 9), as well as our final estimate of the seismic moment, 1.1 xl017Nm. Since the source duration of a Mw = 5.4 earthquake is expected to be about 4 seconds, the details of the source time function cannot be inverted at a period of 5 sec. The only way to obtain some information is through a trial and error analysis using different initial paramctrizations, and knowing that only the low-pass content of the source time function is seen in the inversion. Our best inversion (figure 12) seems to indicate a relatively long source of 5 seconds, similar to the constrained one in figure 8b, and starting at the commonly accepted origin time for this event, lh 20' 2" (vertical bar). This duration reflects the fact that no periods lower than 5 seconds arc included in the inversion. Refining the solution down to 3 seconds, the source appears to have a duration of 4 seconds, more consistent with the source duration obtained at the same period from the moment rates of figure 8a, that is expected to be less affected by structure mis-modelling. But problems of local convergence become more serious at this period, and the global model is more sensitive to the initial one.

The source depth stabilizes at 17.0 kilometers, in agreement with the most recent depth determinations (Table 1). The most notable result is that the cpiccntral location is systematically shifted some kilometers more to the North, placing the main shock at the northern extremity of the aftershock distribution presented by Camclbccck ei at. (1994). A relocation shifted to the South deteriorates the RMS and leads to a regular increase of the strike component, as it can be also seen from the result obtained after application of Silcny's method (up to obtain a pure strike-slip when fixing the location below the 51.100°N meridian). Although we cannot exclude indeterminacies due to the station distribution or to the structure modelling, such a model would imply a directivity effect. In our final tests, we have, therefore, eliminated the point source approximation and we have performed inversions including various models of rupture: along-strikc, anti-strike, and bilateral ones. The corresponding fit of the final synthetic seismograms to the data is shown in figure 13. The visual improvement on the waveform is not very clear and the RMS arc relatively homogeneous (0.24 to 0.29), denoting that the directivity effect for this intermediate magnitude earthquake is not a dominant feature. But the distribution of amplitudes, in particular on the two stations along the rupture direction, ESK and FUR, the best RMS and the correlation coefficient indicate a preference for a unilateral rupture to the South-East. Other directivity models also result in the increase of the strike component and of the seismic moment. This model of rupture to the South-East agrees with the nuclcation point we found at the northern extremity of the aftershock distribution, although the foreshock of the event had been localized at the South-East.

Discussion

We have made an attempt to present a methodology for source inversions at a regional scale, which relies on different, but complementary, schemes for the non linear waveform inversion for the source parameters at the local scale (Silcny and Panza, 1991; Mao el al., 1994), and which have been extended to the regional scale. In the following, we discuss possible improvements or

10 alternatives in the matter of non linear source inversions in areas whose geological structure is well studied. The main problem to take into account is the lateral variation of the structure along the different paths considered. The availability of a structural model such as the I-data set (Du et al., 1997) is important for modelling broadband waveforms at the regional scale, down to periods of some seconds. The methods presented here, however, rely on 1-D, layered structures, Green's functions. In future work it might be desirable to exploit fully the 3-dimcnsionality of the medium provided by the I-data set through the use of 3-D Green's functions. However, when computation time is important, it appears that 1-D averaged structures along each path, or even a unique structure in technically stable regions, can be satisfactorily used. Our tests show that source inversions may require some smoothness in the azimuthal and vertical velocity distributions, and depth-averaging of velocities should be preferred to layered-averages featuring stronger velocity contrasts.

Optimal periods for the inversion have been defined from synthetic tests and arc mostly connected to the distances involved, to the degree of complexity required for the source description, and to the quality of the Green's functions. When using minimal periods of 3 or 5 seconds, we arc able to retrieve with confidence not only the source mechanism, but also the hypocentral location and the source duration. It is found that it is even possible to obtain a better view of the source process than just a point source or a centroid model, since rupture directivity can, and should, be estimated at these periods for M=5.5 and above earthquakes. Shorter periods would be needed to solve the details of the rupture, but they also correspond to the domain of instability of the methods, where the minima of the misfit function become numerous. In particular the source time function, which is the most unstable parameter of source inversions, may be sensitive to many factors such as the structure, the 3-D location, the initial model, and the damping. In fact, the optimal periods of the inversion, for which the minimization of the misfit is clear, arc intrinsically of the same order as the source duration, and just provide information on the low-pass content of the source time function. Body-wave modelling methods (e.g. Nabclck, 1984; Hacsslcr el al., 1992), based on the results of the inversion, might be complementary used to describe the details of the rupture process.

The non linear aspects of the inverse problem have been studied using different iterative approaches, and each one offers its own advantages and disadvantages. Our results show that, whenever a good initial location is not provided, a systematic search within the possible source volume is required to define the location of the best hypoccntcr. The Monte-Carlo random search, complemented with the Simplex method, allows for nice and instructive mappings of the evolution of the function to be minimized over the space of parameters. Genetic algorithms or even an improved Monte-Carlo search, such as the "nearest neighbor, importance sampling" method (Lomax and Snicdcr, 1997) would certainly be effective alternatives because they permit exploration of different local minima, in a self-guided fashion.

11 On the other hand, a simple description of the source mechanism and of the source history at short periods needs several a priori constraints. In particular, the number of inverted parameters should be kept small. Some simple tools of inverse problem analysis, such as the condition number and the resolution, can provide rapid information on the appropriateness of the parametrization, and on the reliability of the result. Introduction of additional a priori correlations may help to optimize the decription of a physical model. A gradient conjugate method which applies separation of parameters proved to be effective for local convergence of the different classes of source parameters, once the starting model has been defined nearby the main global minimum through a more general approach. And the rapidity of such a gradient inversion scheme for a double-couple allows eventually to look for a further directivity effect. Therefore the approaches presented here are complementary and sequential. We first examined the appropriateness of the structural modelling by integrating some degree of freedom in the consideration of the structure and leaving flexibility in the data fitting. Secondly, a thorough exploration of the space of parameters allows for a first general description of the source, which is a suitable starting point for the local retrieval of a simple constrained source model.

Synthetic tests have provided some important indications on the limits of convergence and on the potentialities of non-linear methods at a regional scale. The resolution of the depth is good (i.e. within a few km) at periods of some seconds, while the epicentral relocation is slightly less precise. The mechanism can be safely solved provided a minimal configuration of 4 vertical- component stations offering an azimuthal aperture of 70°. In future developments of the methods, the introduction of SH-Lovc wave data may help stabilizing the results in even poorer configurations. The present inversion of regional data using a sparse network proves that the proposed methodology, the control tools of the inverse problem, and the synthetic tests described here are of major interest for global source inversions in well-mapped regions.

Conclusion

We proposed and discussed here a methodology for the inversion of source parameters at a regional scale, including the focal mechanism, the hypoccntral relocation and the source history. We integrated three aspects: regional structure modelling, systematic exploration of the space of non-linear source parameters using a Monte-Carlo approach, and a local constraining of the solution using a gradient method. Synthetic tests and simple tools of the inverse problem have been used to describe the behaviour of the non-linear problem, and establish the conditions of convergence. The Rocrmond earthquake has been used as a 'test' earthquake to compare these synthetic approaches with real data. It has been shown that such methods, when used with caution owing to

12 the non-linear behaviour of the inverse problem, can provide reliable results even when using a sparse network. Our model shows a normal faulting mechanism identical to those previously published, at a source depth of 17 km. We propose a nuclcation point at the northern extremity of the aftershock distribution presented by Camelbeeck et al. (1994), with a rupture proceeding towards the South-East.

Acknowledgments

During the course of this work, Hugues Dufumier benefitted of a Human Capital and Mobility contract of the European Commission, n° ERBCHBGCT940645. Part of this work belongs to the LP-risk project of the European Commission, contract n° EV5V-CT94-0491. Other fundings include MURTS 40% and 60% funds, the CNR contract n' 96.00318.05 and the European Commission contracts n° EV5V-CT94-0513, ENV4-CT96-0296, ENV4-CT96-0255. We have been using data from the IRIS and Gcofon networks, the MINU1T software (CERN Computer Centre Program Manual 1985), the GMT software (Wessel and Smith, 1991) and the European I-data set (ILP-Task Group 11-4, 3DMET).

13 References

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14 Gciuck M.C., Duin EJ.Th., Dusar M., Rijkcrs R.H.B., van den Berg M.W., and van Rooycn P. (1994): Stratigraphy and tectonics of the Roer Valley graben. Geologic en Mijnbomv, 73, 129-141.

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Loolhuis J. and van Eck T. (1996): Simultaneous focal mechanism and stress tensor inversion using a genetic algorithm. Annales Geophys'tcae, EGS meeting, p. C9I, to be published in Physics and Chemistry of the Earth.

Mao W. J., Panza G.F. and Suhadolc P. (1994): Linearized waveform inversion of local and near- regional events for source mechanism and rupturing processes. Geoph.J. Int., 116, 784-798.

Mao W.J. and Dufumicr H. (1997): Separation of mixed parameters: application to simultaneous inversion of source history, source geometry and hypocentral location. Submitted to Phys. Earth. Plan. Int.

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15 Nformi S., Mao W. and Gubbins D. (1996): Seismic source parameters in New Zealand from broad-band data. Geoph. .7. Int., 124, 289-303.

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17 Table

Reference Latitude, °N Longitude, °E Depth, km Camelbeeckefa/. (1994) 51.163 5.953 17.4 Ahorner(1994) 51.170 5.925 14.6 Pmh&en ctal. (1992) 51.170 5.970 21 Pelzing(1994) 51.168 5.932 17.6 CMT Dziewonski efal. (1993) 51.560 5.630 15 P.D.E. U.S.Geological Survey 51.153 5.798 21 ReNaSS French network 51.150 5.800 - This study, Sileny's method 51.207 5.952 16 This study, Mao's method 51.242 5.938 17.0

Table caption

Table 1. Hypoccntral locations for the April 13, 1992, Rocrmond earthquake: latitude °North, longitude °EasL and depth (km). From previously published studies, Preliminary Determination of Epicentres location from the U.S. Geological Survey, location from the frcnch national network (ReNaSS, WWW source), and the present studies.

18 Figure captions

Figure 1. Distribution of broad-band stations used in this study, around the epicenter of Rocrmond earthquake. The station ESK belongs to the IRIS network while the other ones are from the Geofon network.

Figure 2. 2-D cross-section obtained from the I-data set structure (Du el at., 1997) for the S-wavc velocities along the great circle path Roermond - ESK shown on top.

Figure 3. 1-D structural models used in this study, for each of the six paths of figure 1 and, on the right side, for the source region. The curves represent from left to right the variations of the density p (g/cm3) and of the S-wave and P-wave velocities (km/s) with the depth (km). Dashed lines represent 1-D structures obtained from layer-averaging along the 2-D cross-sections, and solid lines the structures obtained using depth-averaging of the velocities (see text), for the first 100 km depth.

Figure 4. Example of time-frequency analysis used for the processing of the data and of the synthetics, for the path Rocrmond - WET. a) (top) Synthetic phase velocities and group velocities for the first 30 modes of Rayleigh waves corresponding to the 1-D depth-averaged and layer-averaged strucurcs of figure 3. b) (bottom left) Spectrogram, or Frequency-Time-Analysis diagram (Levshin et al., 1992), of the observed scismogram recorded at station WET. The white line shows the limits of the variable period-velocity window selected to filter the data and the synthetics. c) (bottom right) Observed data and Synthetic scismograms, for a normal-fault mechanism at a depth of 20 km, corresponding to the depth-averaged and layer-averaged structures shown on top (fig.4a). The scismograms arc shown before and after application of the variable filtering window shown on the left (fig.4b).

Figure 5. Behaviour of the misfit function at a period of 3 seconds, varying the source depth and a structural parameter Y ranging from 0 to 1. Y=0 represents a unique homogeneous structure for the whole region (the source structure of figure 3), and Y=l particular structures for each source-station path: on figure 5a, the layer-averaged structures of figure 3, and on figure 5b the depth-averaged ones. The grid for the source depth is centered around published depths and extends from 8 to 28 km. Other parameters arc fixed. A value of 1 for the misfit function indicates no minimization of the error, while a value of 0 indicates a perfect fit of the data.

Figure 6. Behaviour of the misfit function over a grid in latitude and longitude of 0.3 x 0.3" covering previously published locations for this earthquake (black dots). The first isolincs arc 0.01, 0.03 and 0.06 units above the absolute minimum. The bold line is the Maas river. Figure 6a shows the minimization at a cutoff period of 3 seconds; figure 6b at 10 seconds.

19 Figure 7. Comparison of the observed data and of the synthetics retrieved from Silcny's method at the best point of the 3-D exploration grid, at a period of 3 seconds. Only the inverted portion of the seismogram is shown, and the number of seconds cut between the origin time and the beginning of the window is indicated on the left, together with the epicentral distance. Each observed seismogram is nonnalized, and its true amplitude is indicated on the right side. The synthetics arc plotted at the same scale as the data. The global correlation between the data and the synthetics is of 0.65, while individual correlations arc indicated on the right.

Figure 8. Moment tensor variations Ml}(i) corresponding to the result of Silcny's inversion at 3 seconds. The instantaneous moment tensor is shown at each time step with an area proportional to its seismic moment. The zero time starts one second before the assumed origin time Ih 20' 2" (vertical bar). The instantaneous moment release is reported below in dotted line, together with the correlated part of the source history, MJi) (solid line), satisfying the positivity constraint. The corresponding average moment tensor M;. is shown on the right, with its seismic moment indicated below. In figure 8a (top) is shown the almost free inversion, using a damping ratio of 0.001, whose average mechanism is used as starting point for the double-couple gradient inversion of part///. In figure 8b (bottom) is shown the a priori constrained solution, using a damping ratio of 0.01, whose average mechanism is also reported in figure 9.

Figure 9. Mechanisms published for Roermond earthquake by other authors and from this study: 1] Camclbccck et ai. (1994), 2] Ahorncr (1994), 3] Paulsscn et ai. (1992), 41 Pclzing (1994), 51 Braunmiller et al. (1994), 6] CMT, Dziewonski et al. (1993), 7] This study, Sileny's method, 8] This study, Mao's method. Cases 11, 21, 31, 41, 81 arc double-couples, cases 51, 61, 71 arc moment tensors. The seismic moment and the strike, dip, rake angles of the best double-couple are indicated below. On case 71, we report the constrained solution from Silcny's method achieving the best correlation fit (0.68) between the radiation of the moment tensor and the polarities observed from broadband and short-period records (black and open circles).

Figure 10. Ellipses of convergence and of error as a function of the mislocation from the true hypocenter, along the vertical and horizontal directions. The scale is given by the wavelength, since the dimensions of these ellipses arc globally proportional to the dominant wavelength used in the inversion. A gradient method converges to the true model (center) when starting from an initial model within the convergence ellipse (within + 1.2 wavelength from the true epicenter and ±2.3 wavelengths from the true depth). While the ellipse of the final errors on the model shows an opposite orientation: the standard error is +0.25 wavelength for the epicentre and +0.15 wavelength for the depth.

Figure 11. Top: Variation of the condition number for the 3 sub-systems solving the source time function (triangles), the mechanism (circles) and the hypocenter (crosses) in Mao's method, for a synthetic case. We present the results as a function of the number of stations, starting from the 6-

20 stations distribution of figure 1 and then eliminating stations clockwise starting from ESK. Bottom: Average number of parameters solved using a damping ratio of 0.001 (i.e. rank of the space of eigenvalues greater than 0.1% of the largest one), for the same sub-systems. The source time function is parametrized here by 10 overlapping triangles, all solved. The 3 parameters describing the mechanism arc not correctly solved when using less than 4 stations, i.e. an azimuthal aperture less than 70°.

Figure 12. Variation of the source model for the Roermond earthquake using Mao's method, at selected iterations. Iteration 0 corresponds to the initial mechanism and location, obtained from Sileny's method (figures 6a and 8a). Iteration 54 is the best one. The function minimized is the RMS fit between the data and the synthetics (as defined by Mao el al., 1994), which is reported on the right. We also indicate below the normalized correlation between the data and the synthetics, and above the condition numbers for the three sub-systems resolving the source time function, the mechanism and the hypocenter. At the bottom of the figure are reported the best source time function and the associated seismic moment, with their error margins. The source time function is parametrized by 5 overlapping triangles of duration 3 s, but only the lowpass content at 5 s of the source time function is seen in the inversion. It was allowed to start one second before the assumed origin time, lh 20' 2", indicated by the vertical bar. The final location is indicated in the lower right corner.

Figure 13. Fit between the data (bottom) and the synthetics for different rupture models. From top to bottom: for the initial model of figure 13, and for the final models obtained considering the point source approximation, a bilateral rupture, a unilateral rupture along the NW strike, and a unilateral rupture along the SE strike. The average RMS error and correlation between the data and the synthetics arc indicated on the right.

21 Roermond , April 13, 1992 350° 10° 20°

60° 60°

50° 50°

40° 40°

350° 0° 10° 20°

22 ESK

\ /'*:

v

,*>••/

ESK Distance (km) Event 600 300 o

! ;• .'•. - -

4.5-

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 S-wave velocity (km/s) Fig.2

23 BFO CLZ ESK FUR HAM WET Roermond Ro Vs Vf)

Fig. 3 WET depth averaging

0,0 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 00 0.1 frequency (Hz) frequency (Hz)

WET layer averaging

0.0 0.1 0.2 0.3 0.4 0.2 0.3 0.5 0.0 0.1 frequency (Hz) frequency (Hz)

-averaging Synth. Layer-overogmg

0 ' 100'200'300'400 0 100 200 300 400 0 100 200 300

t;me (sec) time (sec) time (sec)

Fig. 4

25 Homogeneous to Layer-averaged Structures

Fig.5a

Homogeneous to Depth-averaged Structures

Fig.5b

26 Cost Function Minimization over Epicentral Grid at 3 sec 0.98

Fig.6a

27. Cost Function Minimization over Epicentral Grid at 10 sec 0.65

Fig.6b

28 DATA FIT - Correlation: 0.65

Observed and Synthetic Data

Dt= 79.5s 4.77E-05 BFO Z 362.7 km Cor.: 0.76 "'' 'v V'',.'l' "*\J \i '••••'•

4.65E-05

Cor.: 0.64

1.86E-05

Cor.: 0.42

20.0 40,0 60.0 80.0 100.0 120.0 140.0 TIME (sec)

ig.7

29 MODEL: a time-variable moment tensor

a] almost unconstrained model

AVERAGE

Mo=5.6E+17Nm Mo'(t)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 sec total moment release = 1.0E+18 Nm

b] a priori constrained model

AVERAGE

Mo=1.5E+17Nm Mo'{t)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 sec total moment release = 2.7E+17 Nm

Fig. 8

30 1] Mo=1.4E+17Nm 2] Mo=9.8E+16Nm 4] Mo=5.4E+16Nm 313 20-85/127 70-92 304 22-90/124 68-90 304 20-90/124 70-90 327 22-64/120 70-100

\

5] Mo=9.2E+16Nm 6] Mo=1.3E+17Nm 7] Mo=1.5E+17Nrn 8] Mo=1.1E+17Nm 333 33-78/139 58-98 314 22-98/143 68-87 310 26-114/157 66-79 298 24-101 / 130 66 -85

Fig. 9

limits of convergence (extension of local minimum)

epicenter

final standard error on the model

1 wavelength

»— ••&

31 Model Stability as a function of the station distribution

100 - Q A - A SOURCE TIME FUNCTION

80 O-©MECHANISM CD H I-HYPOCENTER E 60 oc

40 ~ o O

20 -

A A " 3 4 5 Number of Stations

A A A A- A o CO A - A SOURCE TIME FUNCTION (10 parameters) £2 £ G - O MECHANISM (3 parameters) E CO H h HYPOCENTER (3 parameters) CO c Q. b M— o CD

H + -©- -® G ®-"" CD UJ

Fig.II

32 Iter. Source Time Function Mechanism Loc Control

).0km RMS:0.522| o.o Mo=8.1E+16Nm 6.4 29662-140/18455-35 Depth 16.0km Correl. 0.3351

Cond: 371 13 5| 1.5km RMS:0.345| 6.4 29658-133/17652-43 Depth 16.0km Correl. 0.496|

- Cond: 367 13 4| A 6.3km RMS:0.309 29762-109/15333-59 Depth 15.5km Correl. 0.5301

Cond: 410 11 7| Mo 5.5km ; RMS:0.310| t.4 29745-93/12245-87 Depth 17,1km Correl. 0.5541

4.1km 20 Cond: 408165 5| _ RMS:0.261 a4 29830-99/12861 •« Depth 17.0km Correl. 0.566

4.0km 30 Cond: 396 327 5 _ RMS:0.255l a4 29827-101/13064-85 Depth 16.5km Correl. 0.563 4.1 tail| 40 Cond: 386 361 6 RMS:0.251 0.0 Mo=1.0E+17Nm 6,4 29825-101/13065-85 Depth 16.7km Correl. 0.565 4.2krr \ 50 Cond: 391 450 5| RMS:0.250| 29824-101 /130 66 -85 Depth 17.0km Correl. 0.565

4.3krr \ 54 Cond: 391 453 5| Best RMS:0.2451 5,4 29824-101/13066-85 Depth 17.0km Best Correl. 0.569

Best STF + error 0.0 Mo=9.9E+16-1.4E+17Nm Final hypocenter: 51.242 N 5.938 E 17.001 km

Fig.12 33 BFO CLZ WET ESK

5.4471358e-Q5 6.265O295e-05 Cor .335

9.214344Be-G5 1.0J78001e-W 2.3039O61e-05 if.}'.3031967e-05 2.0J66459e-05 RUS .286 —^*|h -——•wm*— wMl Cor .475

8.4457562e-05 -04 2.3435707e-05 .3072667e-O5 2.1306409e-05 Cor .467

1.5924856e-04 ..3301869e-G5 3JI33484e--05 RMS .295 m Cor .387 •I

5.19401846-05 9.7851364e-05 l.3283541e-05 RMS .245 Cor .569

2.7265292e-05 7.B5223096-05 1,iO16O72e-05

Fig.13